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Differentiate w.r.t. t
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3\times 2t^{3-1}+2\left(-27\right)t^{2-1}+108t^{1-1}
The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.
6t^{3-1}+2\left(-27\right)t^{2-1}+108t^{1-1}
Multiply 3 times 2.
6t^{2}+2\left(-27\right)t^{2-1}+108t^{1-1}
Subtract 1 from 3.
6t^{2}-54t^{2-1}+108t^{1-1}
Multiply 2 times -27.
6t^{2}-54t^{1}+108t^{1-1}
Subtract 1 from 2.
6t^{2}-54t^{1}+108t^{0}
Subtract 1 from 1.
6t^{2}-54t+108t^{0}
For any term t, t^{1}=t.
6t^{2}-54t+108\times 1
For any term t except 0, t^{0}=1.
6t^{2}-54t+108
For any term t, t\times 1=t and 1t=t.